{
  "aliases": [
    "rational-root-test"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0EWXPZCSZ0"
    }
  ],
  "body": "\n## Statement\n\nLet $f = a_n x^n + a_{n-1} x^{n-1} + \\dots + a_0 \\in \\mathbb{Z}[x]$ with $n \\ge 1$ and $a_n \\neq 0$.\nLet $r = p/q$ be a rational [[root|00EHX6FWC0]] of $f$, with $p, q \\in \\mathbb{Z}$, $q \u003e 0$ and $\\gcd(p, q) = 1$.\nThen $p$ divides $a_0$ and $q$ divides $a_n$.\n\n## Remarks\n\nThe proof: multiply $f(p/q) = 0$ by $q^n$. Then $a_n p^n = -q(\\dots)$ and $a_0 q^n = -p(\\dots)$, and $\\gcd(p, q) = 1$ gives the two divisibilities.\nFor example, the only possible rational roots of $2x^3 - 3x + 1$ are $\\pm 1$ and $\\pm \\tfrac{1}{2}$. Evaluation shows that $1$ is a root.\nThe same statement holds over every [[unique factorisation domain|06YMXJMXYS]] and its field of fractions.\nFor a monic $f$, every rational root is an integer.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives a finite list of candidates for the rational roots.\nYou test each candidate by evaluation.\nFor $x^3 - 2$, the candidates are $\\pm 1$ and $\\pm 2$, and none is a root, so $\\sqrt[3]{2}$ is irrational.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "rational-root-test"
    ],
    "kind": "statement",
    "msc": [
      "12E05"
    ],
    "name": "Rational root theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q180345",
      "wikipedia": "Rational root theorem"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Rational_root_theorem"
      }
    ],
    "tag": "0819CQ1V8H",
    "uses": [
      "00EHX6FWC0"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0819CQ1V8H/",
  "links": [
    {
      "field": "uses",
      "target": "00EHX6FWC0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0819CQ1V8H.md",
  "msc": [
    "12E05"
  ],
  "name": "Rational root theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/081/0819CQ1V8H.md",
  "status": null,
  "tag": "0819CQ1V8H",
  "uri": "m2://0819CQ1V8H",
  "url": "https://math2.tamnd.com/wiki/rational-root-theorem/",
  "writing": {
    "tag": "0819CQ1V8H",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}